Clustering and classification
# access the packages
library(MASS); library(corrplot); library(tidyr); library(corrplot); library(dplyr); library(ggplot2); ## corrplot 0.84 loaded##
## Attaching package: 'dplyr'## The following object is masked from 'package:MASS':
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## select## The following objects are masked from 'package:stats':
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## filter, lag## The following objects are masked from 'package:base':
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## intersect, setdiff, setequal, union# load the data
data("Boston")
# explore the dataset
dim(Boston)## [1] 506 14str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00Data Set Information
“Boston {MASS}” dataset consist of housing values in suburbs of Boston. The Boston data frame has 506 rows and 14 columns.
This data frame contains the following variables:
crim per capita crime rate by town.
zn proportion of residential land zoned for lots over 25,000 sq.ft.
indus proportion of non-retail business acres per town.
chas Charles River dummy variable (= 1 if tract bounds river; 0 otherwise).
nox nitrogen oxides concentration (parts per 10 million).
rm average number of rooms per dwelling.
age proportion of owner-occupied units built prior to 1940.
dis weighted mean of distances to five Boston employment centres.
rad index of accessibility to radial highways.
tax full-value property-tax rate per $10,000.
ptratio pupil-teacher ratio by town.
black 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town.
lstat lower status of the population (percent).
medv median value of owner-occupied homes in $1000s.
Overview of the data
# Change the shape of the data from wide-format to long-format
require(reshape2)## Loading required package: reshape2##
## Attaching package: 'reshape2'## The following object is masked from 'package:tidyr':
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## smithsmelt.boston <- melt(Boston)## No id variables; using all as measure variableshead(melt.boston)## variable value
## 1 crim 0.00632
## 2 crim 0.02731
## 3 crim 0.02729
## 4 crim 0.03237
## 5 crim 0.06905
## 6 crim 0.02985# draw a bar plot of each variable
ggplot(data = melt.boston, aes(x = value)) + stat_density() + facet_wrap(~variable, scales = "free")
# plot matrix of the Boston dataset variables
pairs(Boston)
# calculate the correlation matrix of the Boston dataset and round it
cor_matrix<-cor(Boston)
# print the correlation matrix
cor_matrix %>% round(digits = 2)## crim zn indus chas nox rm age dis rad tax ptratio
## crim 1.00 -0.20 0.41 -0.06 0.42 -0.22 0.35 -0.38 0.63 0.58 0.29
## zn -0.20 1.00 -0.53 -0.04 -0.52 0.31 -0.57 0.66 -0.31 -0.31 -0.39
## indus 0.41 -0.53 1.00 0.06 0.76 -0.39 0.64 -0.71 0.60 0.72 0.38
## chas -0.06 -0.04 0.06 1.00 0.09 0.09 0.09 -0.10 -0.01 -0.04 -0.12
## nox 0.42 -0.52 0.76 0.09 1.00 -0.30 0.73 -0.77 0.61 0.67 0.19
## rm -0.22 0.31 -0.39 0.09 -0.30 1.00 -0.24 0.21 -0.21 -0.29 -0.36
## age 0.35 -0.57 0.64 0.09 0.73 -0.24 1.00 -0.75 0.46 0.51 0.26
## dis -0.38 0.66 -0.71 -0.10 -0.77 0.21 -0.75 1.00 -0.49 -0.53 -0.23
## rad 0.63 -0.31 0.60 -0.01 0.61 -0.21 0.46 -0.49 1.00 0.91 0.46
## tax 0.58 -0.31 0.72 -0.04 0.67 -0.29 0.51 -0.53 0.91 1.00 0.46
## ptratio 0.29 -0.39 0.38 -0.12 0.19 -0.36 0.26 -0.23 0.46 0.46 1.00
## black -0.39 0.18 -0.36 0.05 -0.38 0.13 -0.27 0.29 -0.44 -0.44 -0.18
## lstat 0.46 -0.41 0.60 -0.05 0.59 -0.61 0.60 -0.50 0.49 0.54 0.37
## medv -0.39 0.36 -0.48 0.18 -0.43 0.70 -0.38 0.25 -0.38 -0.47 -0.51
## black lstat medv
## crim -0.39 0.46 -0.39
## zn 0.18 -0.41 0.36
## indus -0.36 0.60 -0.48
## chas 0.05 -0.05 0.18
## nox -0.38 0.59 -0.43
## rm 0.13 -0.61 0.70
## age -0.27 0.60 -0.38
## dis 0.29 -0.50 0.25
## rad -0.44 0.49 -0.38
## tax -0.44 0.54 -0.47
## ptratio -0.18 0.37 -0.51
## black 1.00 -0.37 0.33
## lstat -0.37 1.00 -0.74
## medv 0.33 -0.74 1.00# visualize the correlation matrix of the dataset
corrplot(cor_matrix, method="number", type='upper', diag = FALSE)
Several of the variables are highly skewed.In particular, crim, zn, chaz, dis, and black are highly skewed. Some of the others appear to have moderate skewness. The skewed distributions suggests that some transformations on variables could improve performance of variables in the models. We can observe several highly correlated variables in the correlation matrix. We have to be careful with highly correlated variables to avoid overcome their influence in the models. The next thing we need to do is standardize the dataset and print out summaries of the scaled data, then create a categorical variable of the crime rate in the Boston dataset using the quantiles as the break points, drop the old crime rate variable from the dataset, and create training and testing data (80% of the data belongs to the train set).
The dataset standardizing and dividing to training and testing datasets
# center and standardize variables
boston_scaled <- scale(Boston)
# summaries of the scaled variables
summary(boston_scaled)## crim zn indus chas
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563 Min. :-0.2723
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723
## Median :-0.390280 Median :-0.48724 Median :-0.2109 Median :-0.2723
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202 Max. : 3.6648
## nox rm age dis
## Min. :-1.4644 Min. :-3.8764 Min. :-2.3331 Min. :-1.2658
## 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049
## Median :-0.1441 Median :-0.1084 Median : 0.3171 Median :-0.2790
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617
## Max. : 2.7296 Max. : 3.5515 Max. : 1.1164 Max. : 3.9566
## rad tax ptratio black
## Min. :-0.9819 Min. :-1.3127 Min. :-2.7047 Min. :-3.9033
## 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049
## Median :-0.5225 Median :-0.4642 Median : 0.2746 Median : 0.3808
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332
## Max. : 1.6596 Max. : 1.7964 Max. : 1.6372 Max. : 0.4406
## lstat medv
## Min. :-1.5296 Min. :-1.9063
## 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 3.5453 Max. : 2.9865# class of the boston_scaled object
class(boston_scaled)## [1] "matrix"# change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)
# summary of the scaled crime rate
summary(boston_scaled$crim)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.419367 -0.410563 -0.390280 0.000000 0.007389 9.924110# create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610# create a categorical variable 'crime'. Using the quantiles as the break points in the categorical variable.
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, label=c("low", "med_low", "med_high", "high"))
# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
# add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)
# number of rows in the Boston dataset
n <- nrow(boston_scaled)
# choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
# create train set
train <- boston_scaled[ind,]
# create test set
test <- boston_scaled[-ind,]
# save the correct classes from test data
correct_classes <- test$crime
# remove the crime variable from test data
test <- dplyr::select(test, -crime)Fit the linear discriminant analysis (LDA) on the train set
Now the test data has created. Next we going to fit the linear discriminant analysis on the train dataset. Notice that in this case we have four classes. The LDA algorithm starts by finding directions that maximize the separation between classes, then use these directions to predict the class of individuals. These directions, called linear discriminants, are a linear combinations of predictor variables.
LDA assumes that predictors are normally distributed (Gaussian distribution) and that the different classes have class-specific means and equal variance/covariance.
LDA determines group means and computes, for each individual, the probability of belonging to the different groups. The individual is then affected to the group with the highest probability score.
The lda() outputs contain the following elements:
Prior probabilities of groups: the proportion of training observations in each group. Group means: Shows the mean of each variable in each group. Coefficients of linear discriminants: Shows the linear combination of predictor variables that are used to form the LDA decision rule.
# linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)
# print the lda.fit object
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2599010 0.2524752 0.2400990 0.2475248
##
## Group means:
## zn indus chas nox rm age
## low 0.9279095 -0.8746763 -0.122344297 -0.8696347 0.4031917 -0.8748606
## med_low -0.1095431 -0.2764984 -0.079333958 -0.5692274 -0.1901812 -0.3339528
## med_high -0.3793843 0.1361663 0.255323541 0.3691752 0.1271016 0.4078957
## high -0.4872402 1.0171519 0.003267949 1.0749693 -0.3366619 0.8258984
## dis rad tax ptratio black lstat
## low 0.8671167 -0.6767071 -0.7197454 -0.43916657 0.38239325 -0.7722497
## med_low 0.3745215 -0.5461293 -0.4782323 -0.05825635 0.35586788 -0.1220811
## med_high -0.3729400 -0.4396053 -0.3562496 -0.27946113 0.05446032 0.0418992
## high -0.8520910 1.6377820 1.5138081 0.78037363 -0.80374216 0.8960484
## medv
## low 0.49700462
## med_low -0.04130269
## med_high 0.17253054
## high -0.71618264
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.05009772 0.64207880 -0.950731976
## indus 0.10356583 -0.10204444 0.337704770
## chas -0.13985022 -0.06768071 0.004420603
## nox 0.33079871 -0.80960156 -1.278168419
## rm -0.17150377 -0.18122296 -0.259021417
## age 0.15039918 -0.33012445 -0.086823868
## dis -0.04219737 -0.21247088 0.218490757
## rad 3.70570914 1.01252252 -0.097687093
## tax 0.03515543 -0.02130031 0.563338138
## ptratio 0.09048200 -0.08257425 -0.303352559
## black -0.09004387 0.06428368 0.164388920
## lstat 0.20576348 -0.37925185 0.341327562
## medv 0.20924378 -0.48027914 -0.152088541
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9582 0.0308 0.0110# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda results
plot(lda.fit, dimen = 2, col = classes, pch = classes)
lda.arrows(lda.fit, myscale = 2)
The train data was devided in quantiles. The crime variable is as actarget variable. In the plot we see four different clusters. Three of them are in overlapped and one cluster is far away from other clusters. Look at the arrows tells us which of the affect most on the classification (rad, zn, nox) but because there is so much variables it is hard to recognize other variables.
Predict the classes with the LDA model on the test data
# predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
## low 16 5 1 0
## med_low 6 13 5 0
## med_high 0 6 20 3
## high 0 0 0 27#Calculate accuracy percent of the model
correct_predicts <- 100 * mean(lda.pred$class==correct_classes)
correct_predicts <- round(correct_predicts, digits = 0)
#Print correct predicts percentage
print(correct_predicts)## [1] 75We split our data earlier so that we have the test set and the correct class labels. The prediction model perform on test data is acceptable but not perfect (prediction accuracy is 75%). It predicts high crime rate perfectly but lower rates worse.
K-means clustering
“Clustering is one of the most common exploratory data analysis technique used to get an intuition about the structure of the data. It can be defined as the task of identifying subgroups in the data such that data points in the same subgroup (cluster) are very similar while data points in different clusters are very different. In other words, we try to find homogeneous subgroups within the data such that data points in each cluster are as similar as possible according to a similarity measure such as euclidean-based distance or correlation-based distance. The decision of which similarity measure to use is application-specific.” (https://towardsdatascience.com/k-means-clustering-algorithm-applications-evaluation-methods-and-drawbacks-aa03e644b48a)
# load the data
data("Boston")
# Standardizing Boston dataset
scaled_boston <- scale(Boston)
# euclidean distance matrix
dist_eu <- dist(scaled_boston)
# look at the summary of the distances
summary(dist_eu)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.1343 3.4625 4.8241 4.9111 6.1863 14.3970# manhattan distance matrix
dist_man <- dist(scaled_boston, method = 'manhattan')
# look at the summary of the distances
summary(dist_man)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618# k-means clustering
km <-kmeans(scaled_boston, centers = 3)
# plot the scaled_oston dataset with clusters
pairs(scaled_boston, col = km$cluster)
set.seed(123)
# determine the number of clusters
k_max <- 10
# calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(scaled_boston, k)$tot.withinss})
# visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
# k-means clustering
km <-kmeans(scaled_boston, centers = 3)
# plot the scaled_boston dataset with clusters
pairs(scaled_boston, col = km$cluster)
I tested many different number of clusters. Based on visualiztion the results suggest that 3 is the optimal number of clusters as it appears to be the bend in the elbow (= when the total WCSS drops radically).
Bonus
# load the data
data("Boston")
# Standardizing Boston dataset
scaled_kmeans_boston <- scale(Boston)
scaled_kmeans_boston <- as.data.frame(scaled_kmeans_boston)
# k-means clustering
km <-kmeans(scaled_kmeans_boston, centers = 3)
lda_kmeans <- lda(km$cluster ~ ., data = scaled_kmeans_boston)
lda_kmeans## Call:
## lda(km$cluster ~ ., data = scaled_kmeans_boston)
##
## Prior probabilities of groups:
## 1 2 3
## 0.2470356 0.3260870 0.4268775
##
## Group means:
## crim zn indus chas nox rm
## 1 -0.3989700 1.2614609 -0.9791535 -0.020354653 -0.8573235 1.0090468
## 2 0.7982270 -0.4872402 1.1186734 0.014005495 1.1351215 -0.4596725
## 3 -0.3788713 -0.3578148 -0.2879024 0.001080671 -0.3709704 -0.2328004
## age dis rad tax ptratio black
## 1 -0.96130713 0.9497716 -0.5867985 -0.6709807 -0.80239137 0.3552363
## 2 0.79930921 -0.8549214 1.2113527 1.2873657 0.59162230 -0.6363367
## 3 -0.05427143 0.1034286 -0.5857564 -0.5951053 0.01241316 0.2805140
## lstat medv
## 1 -0.9571271 1.06668290
## 2 0.8622388 -0.67953738
## 3 -0.1047617 -0.09820229
##
## Coefficients of linear discriminants:
## LD1 LD2
## crim -0.03206338 -0.19094456
## zn 0.02935900 -1.07677218
## indus 0.63347352 -0.09917524
## chas 0.02460719 0.10009606
## nox 1.11749317 -0.75995105
## rm -0.18841682 -0.57360135
## age -0.12983139 0.47226685
## dis 0.04493809 -0.34585958
## rad 0.67004295 -0.08584353
## tax 1.03992455 -0.58075025
## ptratio 0.25864960 -0.02605279
## black -0.01657236 0.01975686
## lstat 0.17365575 -0.41704235
## medv -0.06819126 -0.79098605
##
## Proportion of trace:
## LD1 LD2
## 0.8506 0.1494# the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
# target classes as numeric
classes <- as.numeric(train$crime)
# plot the lda results
plot(lda_kmeans, dimen = 2, col = classes, pch = classes)
lda.arrows(lda_kmeans, myscale = 4)
In the plot we see two overlapped cluster and one cluster which away from other clusters. The arrows tells us thatnox, zn, tax and medv the most influential variables in the model.
Super Bonus
model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutplot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = train$crime)plot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers', color = classes)